Parameter Identification of Weakly Nonlinear Vibration System in Frequency Domain

Shock and Vibration 11 (2004) 685–692 685 IOS Press Parameter identification of weakly nonlinear vibration system in frequency domain Jiehua Penga,b,∗, Jiashi Tanga and Zili Chena,c aDepartment of Mechanics, Hunan University, Changsha, 410082, P.R. China bDepartment of physics, Shaoyang University, Shaoyang, 422000, P.R. China cCollege of Architectural Engineering, Central South Forestry University, Zhuzhou, 412008, P.R. China Received 1 November 2001 Revised 11 December 2003 Abstract. A new method of identifying parameters of nonlinearly vibrating system in frequency domain is presented in this paper. The problems of parameter identification of the nonlinear dynamic system with nonlinear elastic force or nonlinear damping force are discussed. In the method, the mathematic model of parameter identification is frequency response function. Firstly, by means of perturbation method the frequency response function of weakly nonlinear vibration system is derived. Next, a parameter transformation is made and the frequency response function becomes a linear function of the new parameters. Then, based on this function and with the least square method, physical parameters of the system are identified. Finally, the applicability of the proposed technique is confirmed by numerical simulation. Keywords: Parameter identification, weakly nonlinear vibration system, least square method, frequency domain 1. Introduction The parameter identification is one of the important problems of the vibration research. There are many methods of identifying parameters of the linear system. But the parameter identification of nonlinear system is much more difficult than the linear system. Hence, this problem attracts much attention. A. H. Nayfeh has presented an excellent method of parametric identification of nonlinear system that exploits nonlinear resonances and comparisons of the system to be identified with those of known system [1], but the difficult usually appears when we find the known system. Tang et al. [2–4] have suggested time domain method and phase plane method for identifying parameters of nonlinear system which are effective for integrable nonlinear system. K. Yasuda et al. [5,6] have proposed an experiment method of parameter identification which is useful for geometrically nonlinear systems. R. Bachmayer et al. [7] have established adaptive parameter identification method for marine thruster which is an on-line technique for adaptive identification. There are some other methods of parameter identification, such as orthogonal function method [8], wavelet-based technique [9], Volterraseries method [10], cross-correlation analysis [11], power spectrum technique [12], and so on. A new method of identifying parameters of the weakly nonlinear vibration system in the frequency domain is presented in this paper. We discuss two kinds of nonlinear vibration systems with nonlinear elastic force or nonlinear damping force respectively. First, by using the multiple scale method, we obtain the frequency response function of the system which is regarded as a mathematical model of the curve fitting. Next, a new set of parameters are introduced to replace the parameters to be identified and the frequency response function is transformed into the linear function of the new parameters so that the least square method can be used to identify the parameters. This ∗Corresponding author. E-mail: [email protected]. ISSN 1070-9622/04/$17.00 2004 – IOS Press and the authors. All rights reserved 686 J. Peng et al. / Parameter identification of weakly nonlinear vibration system in frequency domain is the key step of our method. Then, with the least square method the normal equations are established and they are linear equations of the new parameters. From the solution of normal equations we can get the values of the parameters to be identified. The method we present is a combination of the perturbation method and the least square method, hence, the method has the same precision with those of perturbation method and least square method. On the other hand, the perturbation method is an effective method to deal with weakly nonlinear system and the least square method is an excellent method of curve fitting such that this parameter identification method has a wide application in the study of weakly nonlinear vibration system. Although frequency response function is the first order approximation of the relationship between amplitude and frequency of nonlinear system, it is the most useful and important mathematical model to study the property of nonlinear vibration system [13]. It can be determined by experiment with good accuracy because in practice any complicated nonlinear system will finally move in a steady motion state if the excitation is harmonic and the system doesn’t come into chaotic motion. Compared with response function in time domain, frequency response function is independent of initial conditions and the test to determine this relationship possesses good repeatability and stability, such that this method of parameter identification can be easily realized by experiment. 2. Principle of the method The governing equation of the vibration system with nonlinear damping force is as follows mx¨ + cx˙ + f(˙x)+kx = G cos Ωt (1) The governing equation of the vibration system with nonlinear elastic force is as follows mx¨ + cx˙ + kx + f(x)=G cos Ωt (2) where f(˙x) is a nonlinear function of x˙, f(x) is a nonlinear function of x. Rewrite Eqs (1) and (2) into the standard form 2 x¨ + ω x = ε[−2nx˙ − f1(˙x)+g cos Ωt] (3) and 2 x¨ + ω x = ε[−2nx˙ − f1(x)+g cos Ωt] (4) k c where ε being a small positive parameter, ω = m being the linear natural frequency of the system, n = 2m , f(˙x) f(x) G f1(˙x)= m , f1(x)= m and g = m . When Ω approaches to ω, we introduce a detuning parameter σ, letting Ω=ω + εσ (5) We can find the approximate frequency response function of the nonlinear system Eqs (3) and (4) by using approximate analytical method, such as the methods of iteration, multiple scales, KBM and so on [13–15]. The general approximate frequency response function of Eqs (3) and (4) can be written in the form F (ω,n,...,a,σ)=0 (6) Where ω,n,..., are physical parameters to be identified of the system Eqs (3) and (4), a is the amplitude, σ is the detuning parameter whose dimension is the same with frequency. Equation (6) plays key role in revealing the properties of nonlinear vibration system and here we take it as a mathematical model of the curve fitting. But F (ω,n,...,a,σ) is a nonlinear function of parameters (ω,n,...,) we must replace the parameters (ω,n,...,) with a new set of parameters (A,B,...,) before employing the least square method to identify the parameters, such that F (ω,n,...,a,σ) can be transformed into a linear function of (A,B,...,). That is, if we chose (ω,n,...,)=R(A,B,...,) (7) then J. Peng et al. / Parameter identification of weakly nonlinear vibration system in frequency domain 687 F (ω,n,...,a,σ)=F1(A,B,...,a,σ)=0 (8) Where F1(A,B,...,a,σ) is a linear function of parameters (A,B,...,). Transform function R(A,B,...,) is determined by the frequency response function of the systems. We use the test data (σ i,ai)(i =1, 2,...,m) for fitting the curve. Substituting σi,ai into Eq. (8) yields small errors Ei = F1(A,B,...,ai,σi) i =1, 2,...,m The sum of the square of all errors is m m 2 2 E = Ei = [F1(A,B,...,ai,σi)] (9) i=1 i=1 It is obvious that the parameters ω,n...will fit to make the totle square errors E get the minimum. This means that we can use the method of the minimum squares to identify the parameters of the system. By determining the extreme value of E and solving a set of normal equations resulted from the extreme value of E, we can identify parameters ω,n,...of the nonlinear system. 3. The system with nonlinear damping Let us consider a weakly nonlinear vibration system with a small cubic nonlinear damping force, its governing equation is as follows. 3 mx¨ + cx˙ + c1x˙ + kx = G cos Ωt (10) Rewrite the standard equation in the form mx¨ + ω2x = ε(−2nx˙ − αx˙ 3 + g cos Ωt) (11) where k g c G ω2 = , 2n = ,α= ,g= (12) m m m m ω,n and α are the physical parameters to be identified of the system Eq. (11). We begin by assuming an expansion of the solution having the form x(t, ε)=x0(T0,T1)+εx1(T0,T1)+... (13) New independent variables are introduced according to n Tn = ε t for n =0, 1,... (14) Substituting Eq. (13) into Eq. (11) and equating the coefficients of ε i, (i =0, 1, 2,...) to zero, we obtain 2 2 D0x0 + ω x0 =0 (15) 2 2 3 D0x1 + ω x1 = −2D0D1x0 − 2nD0x0 − α(D0x0) + g cos(ωT0 + σT1) (16) The solution of the Eq. (15) is x0 = a cos(ωT0 + ϕ) (17) Substituting Eq. (17) into Eq. (16) leads to 1 2 + 2 =2 sin( + )+2 cos( + )+2 sin( + )+ 3 3[3 sin( + ) D0x1 ω x1 ωa ωT0 ϕ ωaβ ωT0 ϕ nωa ωT0 ϕ 4αa ω ωT0 ϕ − sin 3(ωT0 + ϕ)] + g cos(ωT0 + ϕ)cos(σT1 − ϕ) − g sin(ωT0 + ϕ)sin(σT1 − ϕ) To eliminate secular terms from x1, we must put 688 J. Peng et al. / Parameter identification of weakly nonlinear vibration system in frequency domain 3 3 3 2ωa +2naω + 4 αa ω −g sin(σT1 − ϕ)=0 (18) 2ωaϕ + g cos(σT1 − ϕ)=0 Letting σT1 − ϕ = γ (19) Determining the derivatives with respect to T1 yields ϕ = σ − γ . Considering the steady-state response, that is, a =0,γ =0,wehave 2naω + 3 αa3ω3 − g sin γ =0 4 (20) 2ωaσ + g cos γ =0 The result by eliminating γ in Eq.

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